Problem: Maricel is programming an archery component of a new video game. In her code, she has created an "auto aim" feature that helps players more easily hit their intended targets. The code works such that if the player is aimed at $A$ but instead should be aimed at target $T$, the game will automatically adjust the angle at which the arrow is fired directly at $T$. If $A$ and $T$ are $15$ meters apart in the example below, how many degrees will Maricel's code adjust the shot? Do not round during your calculations. Round your final answer to the nearest degree.
Converting the problem into geometrical terms Our problem can be modeled by the following triangle $\triangle ABC$, where we want to find $\angle C=\theta$. $C$ $\theta$ $\;\;\;\;\;\;\;\;\;\;\;\;B$ $A$ $60\text{ m}$ $50\text{ m}$ $15\text{ m}$ Since we are given three side lengths, we can use the law of cosines. Using the law of cosines The law of cosines gives the following equation. $(AB)^2=(AC)^2+(BC)^2-2AC\!\cdot\! BC\!\cdot\!\cos(C)$ Solving the above equation for $\cos(C)$ gives the following equation. $\begin{aligned} \cos(C)&=\dfrac{(AC)^2+(BC)^2-(AB)^2}{2AC\!\cdot\! BC}\\\\ \cos(\theta)&=\dfrac{50^2+60^2-15^2}{2\cdot 50\cdot 60} \gray{\text{Substitute}}\\\\ \cos(\theta)&=\dfrac{5875}{6000}\\\\ \theta&=\cos^{-1}\left(\dfrac{5875}{6000}\right)\\\\ \theta&\approx 12^\circ \end{aligned}$ The answer Maricel's code will adjust the shot by $12^\circ$.